Note: Descriptions are shown in the official language in which they were submitted.
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Determination of Winding/Unwinding Roll Density Using
Time Variant Recursive Techniques
The invention concerns a method for determination of roll density, in
wllic:.li
niethod the density of the web to be wound is detennined in coiuieclion witli
the web reel and which method uses the Icnown relation between the length,
diameter, basis weight and density of the web to be wound
np(D2^Do)
4b
1= length of web to be wound
Tc = 3,1415926...
p = density
D = diameter
Do = diameter of winding core
b = basis weiglit
A calculation foimula is known in the state of the art, whieh cuz be used e.g.
to
detennine the ply thicltness of the wound web or to nieasure the roll density.
The publication Roisum, D.R. "The Measurement of Web Stresses during Roll
Winding", Oklahoma State University, 1990, s. 140 - 141 presents a formula in
connection with measurement of roll density:
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4b (length)
raz-azll '
`C6 GL /1
r
p= average web density over the measured distance
Tc = 3,1415926...
length = cumulated web length over the measured distance
di = roll diameter on measurement occasion i
b basis weight of web
The fonnula was used in Fl patent application 780893 for calculation of the
thickness of a paper web:
d - Rk - Rk-I
nk - nk-1
d(Lk-LkT(Rk -R~2 1),
d = average web thickness over the lneasuring distance
Rk = roll radius on measurement occasion k
Lk = web length on measurement occasion k
7u = 3,1415926...
nk = cumulative number of plies
A similar formula is also presented in the publication Happonen, E. "Paperi-
rullan Rullaustiheyden Mittauslaitteisto" ("Equipment for Measuring tlae
Winding Density of a Paper Roll" Diploma Work), Diplomityo, Teknillinen
Korkeakoulu (University of Technology), 1985 p. 17, which presents a calcula-
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tion formula for the average thickness of wound paper in connection with the
measurement of the thickness of wound paper over a certain winding distance
Dk = (Rk - Rk-1)/(nk - nk-1)=
However, it has proved problematic in the determination of web density or
thiclc-
ness to eliminate the inexactness resulting in measurement results from noise
and
from other disturbances.
As is known in the state of the art, the formula has not been used generally,
be-
cause the density measurement according to the form.ula has been prevented by
the inexact diameter measurement and by the attending vibration frequency of
the
roll center. This is especially problematic when measurement of the location
of
the roll center is used in the diameter measurement.
As regards the state of the art, reference is also made to DE patent
publication
41 28 706, which presents a method for determination of roll tightness when
winding a material web on a winding machine, where the thickness of wound
plies is found out for the roll tightness and wherein the length of the wound
web affecting the diameter is measured directly. In the method, the roll diame-
ter is measured directly by degrees as a transition of the roll center and
then the
ply thickness is calculated using a formula similar to the one described above
(D2 -Di)
4AZ
S average web thickness over the measuring distance
D1= roll diameter in the beginning of the measuring distance
D2 = roll dianieter at the end of the measuring distance
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~c = 3,1415926...
01 web length cumulated over the measuring distance
In the calculation, high-frequency disturbances caused in the diameter meas-
urement by vibration of the roll center are eliminated by low-pass filtration.
In
this known solution the filtration is thus based on the assuinption that
distur-
bances are of a high frequency and the filtration will thus be fairly rough.
The use of low-pass filtration to reduce the share of the noise of the
measureinent
signal is based on the assumption that the noise summed on the measurement is
of
zero average value, that is, unbiased, and that its frequency content differs
from
the measurement signal proper. The filter removes from the measureYnent those
ll.igher frequencies, which the noise brings along, whereby the desired
original
measurement signal will remain. This may cause inexactness, since also a part
of
the desired measurement signal is filtered and also a-part 6f the noise will
remain.
In addition, phase lag, that is, delay, may result in the measurement signal.
If low-pass filtration were to be used in order to achieve an efficient
filtration at
all rnachine running speeds and with all roll diameters, then the boundary fre-
quency of low-pass filtration ought to be changed constantly as these factors
are
changing. Since the main reason for measurement noise is the waving in
diameter
measurement caused by the eccentricity of the roll center, the frequency
content
of measurement noise is strongly dependent on the rotation frequency of the
roll,
which again depends on the running speed and on the roll diameter, being typi-
cally at a maximum a little while after winding has begun. It is a problem
with the
state-of-the-art procedure that it is not easy in practice to implement a
constant
changing during operation of the boundary frequency of any higher rate low-
pass
filter.
Various ways of measuring the roll diameter are known in the state of the art,
the most widely used being pulse measurement, wherein pulse measurement is
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used for measuring the roll circumference. The diameter information obtained
through pulse measurement has been used in order to find out the web thick-
ness. When one ply is completed on the roll, the length measure is taken from
the carrying roller and the paper thickness is obtained by fmding out how
5 quickly the roll diameter increases. It is problematic to determine the web
thickness exactly, because when winding large-diameter rolls wherein the web
is thin, differences in orders of magnitude have caused inexactness.
The roll hardness determines how tightly the roll is wound. This corresponds
with
a certain internal compression pressure distribution between the plies, which
is
the higher the tighter the roll. In addition, roll hardness depends on the
hardness
of the paper itself, that is, on the elastic modulus in the Z direction, which
is dif-
ferent for different paper grades, that is, rolls of various kinds wound to
the same
tightness may have a different hardness. Roll density correlates with
tightness or
1"5 hardiiess, because an iricreased compression pressure will cause a
deformation
that will compress the paper layers together.
The invention aims at bringing about a method more exact than the density meas-
uring methods known in the state of the art. A particular objective of the
inven-
tion is to bring about a method, wherein distortions caused by noise and other
disturbances in the measurement results are eliminated.
In order to achieve the objectives presented above and those emerging
hereinafter
the method according to the invention is mainly characterized in that in the
method distortions caused by noise and by other disturbances in the
measurement
results are eliminated by using in the processing of measurement results a
recur-
sive time variant least squares method.
In the method according to the invention, a value is preferably determined for
"filtration" based on measurement and mathematical statistics and calculus of
probability are preferably applied, whereby an exact measurement results is at-
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tained, when that measurement noise is eliminated, which is mainly caused by
oscillation of the roll center.
In an advantageous embodiment of the method for density measurement accord-
ing to the invention, variables are initialized by two points determined by
the
nominal density before the first measurement. The correlation matrix and
regres-
sion vector are then updated for each following measurement. The correlation
matrix is then reversed, whereupon the paper thickness may be calculated.
Based
on the paper tliickness the density is calculated and the error variance is
updated,
based on which the confidence limit is calculated, for example, for a 95 %
prob-
ability, and if the confidence limit is too big, the buffer length is
increased, while
if the confidence limit is too short the buffer length is decreased. The speed
of
oblivion is then determined and the following measurement is carried out after
a
chosen web length, for example, when the web length has increased by 1 meter.
,_. _ . 15
The method according to the invention uses a recursive time variant least
squares
method, which is easy to implement in program terms and gives the density
value
directly and does not cause any distortions in the shape of the density curve.
To
the least squares method a method of mathematical statistics can be applied,
with
the aid of which a relation is obtained between exactness of measurement and
the
method's built-in filtration constant, whereby statistical confidence limits
are de-
termined for the measurement value, that is, the probability, by which the
meas-
urement value is closer to the correct value by a certain limit. In this
manner any
inaccuracies caused by noise and other disturbances in the measurement results
are controlled in such a way that no distortions will occur in the measurement
results.
Thus, the method according to the invention is based on a statistical method,
which is not dependent on any frequency differentiation of signals, whereby
fil-
tration of measurement results is not performed in a separate stage, but if
required
the effect of filtration is estimated through the confidence limit, the value
of
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which is directly related to the measurement signal proper. In accordance with
an
advantageous feature of the method according to the invention, the program
itself
detennines a suitable filtration constant continuously based on the noise of
the
measurement.
In connection with the method according to the invention the diameter can be
measured as a pulse measurement diameter in connection with winding-in, as a
diameter measured from the location of the roll center, by a distance meter,
for
example, a laser meter or any other suitable measuring procedure that is suffi-
ciently accurate. In unwinding, the location of the roll center does not
change, so
the diameter measurement is most suitably performed as a pulse measurement or
using an ultrasonic distance meter from atop the roll.
In connection with the invention it is advantageous to use a linear sensor in
meas-
uring the tightness of winding-in separately for each station and"based on the
di-
ameter and web length given by the linear sensor.
In the method according to the invention, the diameter may be measured in the
desired manner, for example, by ultrasound, by a laser sensor, using which a
measurement without contact is preferably achieved.
According to an advantageous feature of the method according to the invention,
the determination of the measurement buffer length is also carried out
automati-
cally, whereby the measurement adapts to the varying noise and preserves its
ac-
curacy.
The method according to the invention thus utilizes a physical model in
process-
ing the measurement. When using a least squares procedure in this method ac-
cording to the invention, this advantage is obtained, that is, an unbiased
estimate
of the density is obtained from the measurement data.
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According to an advantageous additional feature of the method according to the
invention, the measurement buffers are initialized in such a way that the meas-
urement first shows the density value given by the user or the density value
measured in the beginning of the previous winding, whence it begins following
the measured value, as the roll starts rotating. This speeds up penetration of
the
initial transient and the measurement is made to begin as early as possible.
An even more exact initialization is achieved by first storing a suitable
quantity
of measurement data and by calculating the density backwards towards the
smaller diameter from this data, whereby a very exact initial value is
obtained
for the roll bottom. However, this is tougher in terms of calculation.
In accordance with an advantageous application of the invention, the measur-
ing method according to the invention may be extended fiuther by applying an
15-- Extended time variant Kalman Filter. The purpose of-the KalmairFilter is
to -- -
utilize not only the direct measurement but also a known or estimated dynamic
model of the system, which includes a physical description of the system
known beforehand:
x = f (x(t),u(t),t)+ w(t)
z(t) = h(x(t),u(t),t)+v(t)
In this description x is the state vector, which defines the interior state of
the
system. In a winding model components of the state vector are e.g. the radial
compression pressure and tangential tensile stress of the roll's surface ply.
Vector u includes control magnitudes, such as e.g. the nip load, web tension,
winding power, running speed. The control magnitudes are lcnown or they can
be measured directly. Vectors w and v are noises disturbing the system and the
measurement. Vector z is a measurement magnitude, in this case the density.
Function f is a description of the winding model, it tells how the roll's
internal
stress distribution results from the effect of the control magnitudes (for
exam-
ple, Jorkaina, M. "Contact Mechanical Model for Winding Nip"; Teknillinen
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Korkeakoulu, 2001 and Hakiel, Z. "Nonlinear Model for Wound Roll Stress" ;
Tappi Journal, 1987). The explicit time variable tells the effect on it by mag-
nitudes not mentioned separately, such as e.g. the roll diameter and the paper
thickness of the arriving web. Function h is a description of pressures
through
roll defonnation on the density. It is dependent on the paper characteristics,
such as the radial and tangential elastic modulus and the friction. The model
may also be static, if dynamics are of no significance (slow changes of
ratings):
x(t) = f (u(t),t)+w(t)
The idea of the Kalman filter is to form a reverse description h-1, which is
used to measure the roll pressure distribution indirectly, when the measured
density and the dynamic physical model describing the winding are known.
Forming the extended time variant Kalman filter is known in the art beginning
from the system description. The model structure presented herein is by no
means the only one that can be applied to winding, nor is the intention to be
limited to it only.
The method according to the invention gives by little loading of the processor
and
light calculation and with easy iinplementation in a program such a result in
an
optimum manner in terms of mathematical statistics, which is very easy to set
up
for use with different paper grades.
In the following, the invention will be described in greater detail with
reference to
the figures shown in the appended drawing, wherein
Figures 1A and 1B are schematic block views of the density measurement ac-
cording to the invention, and
Figure 2 is a schematic view of an example comparing the previously known
pulse density measurement and the measuring method according to the invention.
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The variables shown in the fonnulas in Figures 1A and 1B have the following
meanings:
Do = initial diameter c~ = regression vector =[D21]T
5 D = measured roll diameter Y measurement =1
1= measured web length in roll 0 parameter vector, whose first
b = basis weight given from screen component is inversely proportional to
p = nominal density given from screen the paper thickness
X = oblivion parameter (<1)
In the advantageous application of the density measurement method according to
the invention shown in Figures lA and 1B before the first measurement
occasion,
measurement occasions, block 11, the variables are initialized by two points
de-
termined by the nominal density, block 12, formula:
n.,o15 l - (D2 -Do )
4b
Thereafter, on each measurement occasion the correlation matrix and the regres-
sion vector are updated, blocks 13, 14, formulas:
14 Z
{ z
R~~ = ~~~'1 = Z~~+1 ~n+l + Y'n Y'i~= ARn_~ D D
D2l [.D 1
Y~Yn - AY'Yn-1 +'l~nYn - A~Kt-1 +
Thereafter, the correlation matrix is reversed, whereupon the paper thickness
can
be calculated, blocks 15, 16, formulas:
Pn = R,I 1= 1 R22 -R12
det R -R12 Rl l
(Bn - PnOYn )
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papeY _ thickness = ~
4(1'ti0Yi +1'12OY2)
Based on the paper thickness, the density is calculated, block 17, formula:
Density=Basis_Weiglat/Paper-Thickness
and the error variance is updated, block 18, formula:
sõ = K - 8" 0.
EnronVanianceõ = a= EnnonVaniance,t-I +(l - a)
in which context the confidence limit is calculated, for example, for a
probability
of 95 %, block 19, formula:
ConfidenceLirnitõ =1.739606726 *Basis Weight 4 EnrorVarianceõP,
and if the confidence limit is too large, the buffer length is increased,
whereas if
the confidence limit is too short, the buffer length is decreased, blocks 20,
21.
Thereafter the speed of oblivion is determined, block 22, formula:
~ 1- BufferLength
BufferLength
and the following measurement is performed after a chosen web length, for ex-
ample, when the web length has increased by 1 meter, block 23.
Figure 2 is a schematic view of an exam.ple comparing the previously known
pulse density measurement and the measuring method according to the invention.
In the figure, reference number 31 indicates the curve achieved with pulse
density
measurement and reference number 32 indicates the result curve achieved with
the density measurement according to the method in accordance with the inven-
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tion. As can be seen in Figure 2, the method according to the invention, gives
an
exacter and more reliable result than the previously known density measurement
method.
In the foregoing, the invention was described by referring to its one
advantageous
application example only, but the intention is not to limit the intention in
aily way
strictly to the details of that example.
